Download Boolean Algebra AND Function

Boolean Algebra
Learning Objectives
• Learn that Boolean algebra produces a result that
equals TRUE or FALSE
• Learn how truth tables are used to represent
Boolean expressions
• Learn how to use the AND, OR and NOT operators
on their own or grouped together.
• Learn how to use NAND, NOR and XOR operators
• Learn how to simplify Boolean expresssions
• Learn how De Morgan’s Law allows Boolean
expressions to be created using only NAND or NOR
operators.
Boolean Algebra
•Computers work essentially on logical instructions.
• One way of demonstrating this is with Boolean Algebra and visually with
Logic gates.
A shorthand notation for a system of logic originally set forth by Aristotle and
developed by George Boole (Boolean) is considering a set of statements
either true or false.
It is
raining
today
A statement
Possible Outcomes
True
False
If the statement is represented by the variable X then you can say
X
Meaning
True
It is raining today
False
It is not raining
today
or
X
Meaning
1
It is raining today
0
It is not raining
today
Boolean Algebra
This is Boolean algebra – the concept of whether something is true or false.
Since the introduction of digital electronics and computers which works with
binary – a state of 0s and 1s – Boolean Algebra has now become very useful.
Boolean Algebra can be represented in Truth Tables.
Take this electronic diagram
Boolean Algebra
Let transfer that diagram into a truth table.
Switch
Lamp
Lamp
Switch
Closed
Lamp
Off
Switch
Closed
Open
Closed
On
False
False
0
0
True
True
1
1
OR Function
The above was just stating that if this state is true then this happens. Now
we are going to say if this is true OR this is true then this happens.
Consider the following diagram.
In programming terms this would be something like:
IF (num1 < 10 || num1 > 20) ….
Boolean Algebra
X
Y
Lamp
X
Y
Lamp
Open
Open
Off
0
0
0
Open
Closed
On
0
1
1
Closed
Open
On
1
0
1
Closed
Closed
On
1
1
1
Boolean Algebra
This can be written in equation form
X +Y = Q
Where Q is a true setting for Lamp on. The + here means OR and not addition
like in normal arithmetic.
A truth table for the above will be the same as previous except that you
replace the Lamp with Q.
X
Y
Q
0
0
0
0
1
1
1
0
1
1
1
1
This can be shown in a black box
diagram as:
X
Y
OR
Q
Boolean Algebra
AND Function
Consider the following diagram
In this diagram both X and Y need to on or true for the lamp to be on. This is
like saying that your fridge needs to be switched on and the door open for the
light to come on.
The equation syntax for this is to use a dot. So if Q is the lamp again.
X.Y = Q
And the truth table would look like this –
In Programming terms:
IF (num1 > 10 && num1 < 20) …
X
Y
Q
0
0
0
0
1
0
1
0
0
1
1
1
Boolean Algebra
NOT Function
The not function can be stated as such:
If a State P was true then R is false and vice versa. This can be put in a truth
table like so:
P
R
0
1
1
0
From this we can state the P = NOT R – or the opposite of what R is. The
syntax for this in equation form is this:
_
P=R
(note the _ over the R)
Programming terms:
IF (!valid) ….
Boolean Algebra
Combinations of AND and OR
You can use more complex Boolean algebra expressions by combining AND
and ORs together. Consider the following diagram
The notation for this can be written as:
Q = (X AND Y) OR (X AND Z) or alternatively be written as:
Q = X.Y + X.Z or Q = X.(Y + Z)
Create a truth table for the above diagram.
Boolean Algebra
NAND Function
This is a combination of a NOT and an AND
The notation would be something like this
A .B
Can you draw a truth table for a NAND Function
Boolean Algebra
NOR Function
This is a combination of a NOT and a OR
The notation would be something like this
A+B
Can you draw a truth table for a NOR Function
Boolean Algebra
XOR Function
This function is true only if one value is true and not both.
The notation would be something like this
A
B
Can you draw a truth table for a NOR Function
Boolean Algebra
Simplifying Boolean Expressions
Some complex Boolean expressions can be simplified. This will
enable circuitry to also be simplified in a computer.
For instance Draw a truth table for the expression: (A . B) + A
What do you think this can be simplified to?
Task
Use truth tables to show that the laws on table 30.12 on
page 251 are indeed the case. Put your answer in Specification
Journal 19.
Boolean Algebra
De Morgans Law
This law is another way of simplifying Boolean equasions.
1) In a single expression invert all the ANDs to ORs and vice
versa
2) Invert the whole expression – from NOT to normal and vice
versa
3) Rule 1 – NOT (A AND B) is the same as (NOT A) OR (NOT B)
4) Rule 2 – NOT (A OR B) is the same as (NOT A) AND (NOT B)
WORKING EXAMPLE
Boolean Algebra
De Morgans Law
Task
Answer Question 4 on page 255 of the Text Book.